It was standing room only as Marcus du Sautoy and Ben Okri discussed the relationship between narrative and proof at the Mathematical Institute on Tuesday evening (20 January).
The full video of the event to launch The Oxford Research Centre in the Humanities' (TORCH) 'Humanities and Science programme can be seen here.
The speakers have given Arts Blog permission to publish edited extracts of their talks:
Marcus du Sautoy, Simonyi Professor for the Public Understanding of Science, Oxford University
In Borges's short story The Library of Babel the librarian who narrates the story begins with a description of his place of work:
The universe (which others call the library) is composed of an indefinite and perhaps infinite number of hexagonal galleries...From any of the hexagons one can see, interminably, the upper and lower floors.
As befits a library, this vast beehive of rooms is full of books. The tomes all have the same dimensions. 410 pages, each page with 40 lines and each line consisting of 80 orthographical symbols of which there are 25 in number.
As the librarian explores the contents of his library he finds that most of the books are formless and chaotic in nature but every now and again something interesting appears. He discovers a book with the letters MCV repeated from the first line to the last. In another, the cacophony of letters is interrupted on the penultimate page by the phrase Oh time thy pyramids and then continues its meaningless noise.
The challenge the librarian sets himself is to determine whether the library is in fact infinite or, if not, what shape it has. As the story develops a hypothesis about the library is proposed.
The Library is total ... its shelves register all the possible combinations of the twenty-‐odd orthographical symbols (a number which, though extremely large is not infinite): in other words, all that it is given to express, in all languages. Everything.
The library contains every book that it is possible to write. When it was proclaimed that the Library contained all books, the first impression was one of extravagant happiness. But this was followed by an excessive depression. Because it was realized that this library that contained everything in fact contained nothing.
So what is in the mathematican’s library? I think many believe that it is aspiring to be a mathematical Library of Babel. That the role of the mathematicians is to is to document all true statements about numbers and geometry. The irrationality of the square root of 2. A list of the finite simple groups. The formula for the volume of a sphere. The identification of the brachistochrone as the curve of fastest descent.
Mathematics though is very different from simply a list of all the true statements we can discover about number. Mathematicians, like Borges, are story tellers. Our characters are numbers and geometries. Our narratives are the proofs we create about these characters.
Let me quote one of my mathematical heroes Henri Poincaré articulating what it means to do mathematics:
To create consists precisely in not making useless combinations. Creation is discernment, choice...The sterile combinations do not even present themselves to the mind of the creator.
Mathematics, just like literature, is about making choices. What then are the criterion for a piece of mathematics making it into the journals that occupy our mathematical library? Why is Fermat’s Last Theorem regarded as one of the great mathematical opus’s of the last century while an equally complicated numerical calculation is regarded as mundane and uninteresting. After all, what is so interesting about knowing that an equation like xn+yn=zn has no whole number solutions when n>2.
What I want to propose is that it is the nature of the proof of this Theorem that elevates this true statement about numbers to the status of something deserving its place in the pantheon of mathematics. And that the quality of a good proof is one that has many things in common with act of great story telling.
My conjecture if I was to put it into a mathematical equation is that
"proof = narrative."
A proof is like the mathematician’s travelogue. A successful proof is like a set of signposts that allow all subsequent mathematicians to make the same journey. Readers of the proof will experience the same exciting realization as its author that this path allows them to reach the distant peak.
Very often a proof will not seek to dot every i and cross every t, just as a story does not present every detail of a character’s life. It is a description of the journey and not necessarily the re-enactment of every step. The arguments that mathematicians provide as proofs are designed to create a rush in the mind of the reader. The mathematician GH Hardy described the arguments we give as 'gas, rhetorical flourishes designed to affect the psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils'.
What is important for me about a piece of mathematics is not the QED or final result but the journey that I’ve been taken on to get to that point, just as a piece of music is not about the final chord. It is certainly important to know that there are infinitely many primes but the satisfaction comes from understanding why.
The joy of reading and creating mathematics comes from the exciting “aha” moment we experience when all the strands seem to come together to resolve the mathematical mystery. It is like the moment of harmonic resolution in a piece of music or the revelation of who-‐dunnit in a murder mystery.
The element of surprise is an important quality of exciting mathematics. Here is mathematician Michael Atiyah talking about the qualities of mathematics that he enjoys:
I like to be surprised. The argument thatfollows a standard path, with few new features, is dull and unexciting. I like the unexpected, a new point of view, a link with other areas, a twist in the tail.
When I am creating a new piece of mathematics the choices I will make will be motivated by the desire to take my audience on an interesting mathematical journey full of twists and turns and surprises. I want to tease an audience with the challenge of why two seemingly unconnected mathematical characters should have anything to do with each other. And then as the proof unfolds there is a gradual realization or sudden moment of recognition that these two ideas are actually one and the same character.
The importance of the journey to mathematics can be illustrated by a strange reaction that many mathematicians have when a great mathematical theorem is finally proved. Just as there is a sense of sadness when you come out the other side of a great novel, the closure of a mathematical quest can have its own sense of melancholy.
I think that we’d been so enjoying the journey that Fermat’s equations had taken us on that there was a sense of depression that was mixed with the elation that greeted Andrew Wiles’s solution of this 350 year old enigma. That is why proofs that open up the ground for new stories are valued very highly in mathematics.
Ben Okri, author of Flowers and Shadows, The Famished Road (Booker Prize winner 1991), Songs of Enchantment and Infinite Riches
Narrative is woven into the fabric of consciousness as mathematics is woven into the fabric of the world. If it were possible to imagine a consciousness at the heart of all things in stone, in the air, in trees, in mountains, in stars in atoms, in all things that constitute reality, that consciousness would perceive the world in its smallness, as well as in its largeness, as a grand perpetual narrative.
The motion of things is the story of things; the constant change that Heraclitus saw, the mutations the compositions, visions, collisions, growths, deaths, fissions are all part of the infinite narrative that is reality. It would seem that story is implicated in the world, on the condition that there is a consciousness to perceive it. To that degree narrative is woven into the fabric of the mind. Is there a mathematical basis to narration? Is there a mathematics of narration?
We see from Marcus’s excellent paper that there could be said to be a link between proof and narration, that to prove is to narrate. In literature narrative is a kind of proof. It is more than that of course, but it is always a kind of proof. It is the out figuring of an intuition of tension, of the need for psychic resolution, the need to make visible in order to make understandable.
But narrative as proof takes a more intangible, more aesthetic form. The desire to write a story is not merely to prove the existence of the story in the mind, for often the story does not exist until it is written; it exists in an ideal state, as a throb as Nabakov would call it, an impulse, a pressure on the literary glands, an ache in the soul. But as something cannot come from nothing, the existence of story is proof in a way of a previous intangible condition, a sediment, a concretisation of an opalescent state.
There is one way in which narrative shares a profound similarity to mathematics and this in the unavoidable logic of storytelling. The equation must work. Where the story begins, how it evolves, where it goes, must work, it must add up, it must compute. There is mathematics in narration, in the sense that we know when a story's mathematics does not work. This can happen in a quantum sense at the level of sentences, or in a larger sense in terms of the whole. Narration conforms to an aesthetic mathematics, and the best storytellers therefore are profound and rigorous thinkers. To work out the inner maths of a story or a novel is one of the most difficult things that writers do. For on the rightness of that maths, the unfolding of character within the limitation of plot, rests the immortality or oblivion of the text.
Mathematics is not just what mathematicians do, mathematics – the relation of number – is implicated in the structure of reality, in the number of vibrations that make the atom in the periodicity of elements, and the pulsing of quasars and the calyxes of flowers, in the rhythm of all things. In fact it is the relationship between the rhythmic quality of the world and the rhythmic quality of art that so fascinated Leonardo da Vinci. It would seem that the world, its underlying structure, is governed by known and unknown laws of mathematics. That, as Pythagorus implied, number governs the world, the materiality of the world, the manifestation of things.
It is now axiomatic in practical science that by altering the number of the vibrations of a thing you can alter its nature. I have found this to be true in narration as well. There is a novel of mine called Astonishing the Gods, whose nature was changed by altering its underlying beat, its vibration. I will give you an example. In the early draft it began "invisibility is best". The novel was written in that contracted beat. When I came to write it and to re-write the rewriting I realised that there was something not right about that beat for that novel. In the final version it reads "It is better to be invisible".
A great difference in the unfolding of the text emerged from altering the microbeat of the novel. You could call this re-writing but I believe it is something more. I believe that everything exists by virtue of the law of numbers, its underlying vibration. The elements have their rates of vibration, at atomic and subatomic level all is number, all is vibration, this too at the larger levels. The world is what it is by virtue of this quality. This is beyond the scope of our present conversation, but one can conjecture there might also be a sublime mathematics and that someday could reveal that which we now see as transcendent.
Mathematics is woven into the fabric of the world, narrative into the fabric of consciousness; they are both part of the fact of reality. The world exists by virtue of numbers and the fact that the world exists already implies its narrative quality. The curious thing about both is it they need an intelligent consciousness to perceive their existence.
But there are fundamental differences between proof and narration. One is the universality of language, the transparency of narration. Another is the dual nature of narration that it is both of time and timeless. Another is the mimetic quality of narration, that it mirrors the world, it mirrors modes of consciousness, and in its imaginative dimension it contains matrixes of the future.
One could say that the purity of mathematics makes it more akin to music. Both create realms of their own, crystalline and pure, immaculate fantasies with unalterable laws. Another crucial difference is the fact that great narration has higher immeasurable symmetries. The difference between a well plotted detective novel and Hamlet is infinite resonance. This is not a disquisition on lowbrow and highbrow. A murder mystery of Agatha Christie, for example, is satisfying in tying up all the loose ends, but the resolution of Oedipus Rex gives us unending insight into the complexity of fate, the human condition, and the moral law.
The refractions of Oedipus Rex resonates 2500 years later and the chief reason, I believe, is that it lives, and it partakes of higher truths that we still have not fully grasped. And it compels us to contemplate all the dimensions of what it means to be human. This brings me to the greatest difference between narrative and proof, between mathematics and narration. Narration is, at its highest, about the enigma of being human, of being alive, of consciousness itself, in all of its states, in all its realities, in all its unrealities.
Narration is the highest tool the human mind has devised to investigate the mystery of life and the human condition. It is more than a mirror of the world and ourselves. It is more than an order perceived in the chaos. It is a technology we have dreamt up to help us get from our darkness to our progressive light. I believe that the oldest technology is not the wheel, not the discovery of fire, not even the discovery of language itself. I believe it is storytelling.
We tell stories even without language. We tell stories with our faces, in our gestures, in our eyes. We tell stories on cave walls. I believe the impulse of narration has led us to language and not the other way around. Language amplifies the power and scope of narration. Narration is implied in germination, procreation and cessation. Narration begins before birth and continues after death. Archaeology confirms the latter. Ovid’s Metamorphosis hints at the fact of change which is at the heart of narration.
The big bang was an event of incalculable mathematical magnitude, but it was also a singular narrative event. It could also be called the big beginning. In that moment of astonishing singularity was born the mathematics of all things and the narrative of all things - two children of the same mother-father moment, that brings us all here today.
'Narrative and Proof' was held at the Mathematical Institute on 20 January 2015 at 5pm. Sir Roger Penrose and Professor Laura Marcus also spoke at the event, which was chaired by Professor Elleke Boehmer.